The Complexity of Dynamic/Periodic
Languages and Optimization
Problems
James B. Orlin
Sloan W.P.
No.
1679-85
July 1985
2
1. Introduction
Deterministic dynamic/periodic optimization problems arise naturally in
various quantitative disciplines including Computer Science, Economics, and
Operations Research. These periodic models may be applied to long-range
economic planning, workforce scheduling, vehicle routing, machine maintenance,
and a host of industrial applications. In this paper we offer a unifying
framework for dynamic/periodic problems in
languages",
"terms of dynamic/periodic
and we discuss the complexity of these languages.
In particular,
many such languages derived from NP-complete languages can be shown to be
polynomial space (PSPACE) complete. Among these are the dynamic/periodic
variants of the following problems: the 3-satisfiability problem, the 3dimensional matching problem, the number partition problem, the hamiltonian
path problem, and the independent set problem. We provide a straightforward
technique for showing how to prove the PSPACE-completeness of these periodic
problems derived from NP-complete problems.
PSPACE may be characterized in a number of different ways. Chandra and
Stockmeyer [CS] and Schaefer [Sc] showed that problems in PSPACE could be
reduced to certain 2 person games. More recently, Papadimitriou [P] showed
that the class PSPACE could be described in terms of certain generalizations
of Markov decision chains, or as he phrased it, "games against nature". Our
approach here is to describe the class PSPACE as those languages for which
there are "periodic certificates".
This interpretation lends itself well to proving that many NP-complete
languages have dynamic/periodic counterparts which are PSPACE-complete.
Moreover, these dynamic/periodic languages arise naturally via periodic
optimization problems.
In addition, Reif
R] showed that a number of other dynamic problems are
PSPACE-complete. Although his class is
different from the class described
here, his approach is conceptually similar to ours and his proof technique is
similar.
The results in
this paper were presented in
a preliminary form in
[01].
The Outline of This Paper
In Section 2, we review some of the research on various dynamic/periodic
problems. In Section 3 we illustrate how to derive periodic problems from
problems in the class NP.
3
In Section 4, we define "dynamic/periodic languages" and prove some
elementary properties of these languages. In Section 5, we prove the PSPACEcompleteness of certain periodic languages derived from NP-complete languages.
These results contrast with those by Graves and Orlin [GO] and by Orlin [02],
[03],
04], [05] and [06] who provide polynomial time recognition algorithms
for various dynamic/periodic languages derived from polynomial time languages,
including the following: linear programming, network flows, coloring interval
graphs, maximum matching,
and 2-satisfiability.
In section 7, we discuss the real-time, complexity of dynamic/periodic
optimization problems. Here, periodic optimization problems are viewed as
real-time scheduling problems with known periodic data. We show that there is
a polynomial real-time scheduling algorithm for the periodic optimization
problems presented in Section 3 if and only if P - PSPACE.
In Section 6, we show that the periodic knapsack problem may be solved in
pseudo-polynomial time (i.e., in polynomial time if the data are unary
encoded), and thus it is meaningful to describe other PSPACE problems as
"strongly PSPACE complete", a counterpart to strong NP-completeness as
described by Garey and Johnson [GJ]. Finally, in Section 8, we present some
variations of dynamic/periodic language problems.
2. Some Periodic Optimization Problems
Dynamic/periodic optimization problems occur in a variety of settings.
Below we reference some examples of such optimization problems. The list below
is
intended as a sampler and is far from comprehensive.
Cyclic Staffing
Two types of periodic optimization problems arise typically in
employment: shift scheduling and day-off scheduling. Both of these are
surveyed by Baker [B]. Many problems of this type arise in service industries,
and they may be classified according to the following parameters: types of
workshifts allowed, the demand structures, and the types of additional
constraints, possibly imposed by union contracts.
Vehicle Routing
Dantzig, Blattner, and Rao [D] consider the problem of determining an
infinite horizon tour for a tramp steamer so as to maximize its average profit
per day. The tramp steamer is required to travel among a fixed finite set of
4
ports and receives a profit Pij each time that it travels from port i to port
j. The above problem, also called the minimum cost-to-time ratio circuit
problem, deals with a transportation problem in which the opportunities for
profit are periodic over time. Efficient algorithms for the tramp steamer
problem have been given by Dantzig et.al [D], Lawler Ll, and Megiddo [M].
Another setting for periodic problems is the area of airplane scheduling
(scheduling vehicles to meet a daily repeating set of demands) as surveyed by
Simpson [Si].
Scheduling Tasks on Processors
Many job shop problems may be formulated in an environment in which the
tasks arrive periodically over time, and the task due-dates also repeat
periodically. Subclasses of these problems have been studied by many authors
including Labetoulle [La], Dhall and Liu [DL], Lawler and Martel [L], and Mok
[Mo].
Continuous Problems
An entire literature exists on periodic problems in control with
applications to many industrial settings,- as described in detail in [Ma].
Furthermore, discrete approximations of these models are, in general, special
cases of the class of periodic languages defined in Section 4.
3. Dynamic/Periodic Languages Derived from Languages in the Class NP
In this section we describe dynamic/periodic languages that may be
considered extensions of languages in the class NP. All of the languages below
can be proved to be PSPACE-hard using the technique illustrated in Section 5.
These problems also serve to motivate the definitions and theorems of Section
4 on dynamic/periodic languages.
Notation
We let Z denote the set of integers. We let 1 denote a row vector each of
whose components has value one. The notation .1 refers to the sup norm.
In general, we will let superscripts denote a time dimension. If
S
{s, l .. , .
s n}
is a finite set of elements then
5
S
-
sP: i
[l,,n
and p
Z).
We interpret the element si as the element si in period p. We will also refer
to sP as the p-th copy of element si. The successor of element sP is s+l and
is denoted SUC(sP). If T c S is a finite subset, then the successor of T is
the subset of S obtained by replacing each element of T by its successor. We
let the successor of T be denoted SUC(T).
Rather than use the cumbersome prefix "dynamic/periodic" throughout this
paper, we will typically use the abbreviation D/P.
BOUNDED D/P INTEGER PROGRAMMING
INPUT: m x n matrices A and B, m-vector d, scalar k.
QUESTION: Is there an infinite sequence {x : i
integer n-vectors such that Ixil < k and Axi +
Bxi + !
Z} of non-negative
- d for all i
Z?
DIP PARTITION
INPUT: n-vectors a and b.
QUESTION: Is there an infinite sequence {xi: i
such that ax
+
Z} of boolean n-vectors
+ bx i - d (where d - (la+lb)/2) for all i e Z?
Suppose that U -
ul,i,..., Un,
n}
is a finite set of literals. We say
that a set C of clauses defined on U is periodic if the following property
holds: c
C if and only if SUC(c)
C. (Here the successor of c is obtained
by replacing each literal of c by its successor in U ). We note that to
specify the input for a periodic set of clauses, it suffices to list those
clauses whose initial literal is in period 1.
D/P SATISFIABILITY
INPUT: A finite set U of literals and a periodic collection C of clauses
defined on U .
QUESTION: Is there a truth assignment for all variables such that C is
satisfied?
One can carry the analogy to NP-complete problems further. For example,
one can define D/P 3-Satisfiability to be special case of D/P Satisfiability
in which each clause has exactly 3 literals.
6
PERIODIC GRAPH THEORETIC LANGUAGES
Let V - {1,...,nl be a finite set of vertices. A set E of edges on
is
said to be periodic if it has the following property: (iP,jr)
E if and only
r
+
if (iP+l,Jf
) E. (Equivalently, e
E iff SUC(e)
E.) In addition, there
are a finite number of edges incident to each vertex.
An example of a periodic graph is given in Figure 1. To specify the input
of a periodic graph it suffices to specify those edges whose tail is incident
to a vertex in period 1.
[INSERT FIGURE 1 HERE]
Given the richness of graph theoretic applications, it is not surprising
that periodic graphs are useful descriptors of dynamic/periodic problems.
For
several applications of periodic graphs see [GO], 1021 and [05].
Periodic graphs are an infinite extension of time expanded networks.
These networks are well known and have been used, for example, by Ford and
Fulkerson [FF] in their work as dynamic network flows.
PERIODIC HAMILTONIAN PATH
INPUT: A periodic graph G - (V ,E)
QUESTION: Is there a path P in G that passes through each vertex exactly
once?
PERIODIC 3-COLORABILITY
INPUT: A periodic graph G - (V ,E).
QUESTION: Is it
possible to color all of the vertices of G with only
three colors such that adjacent vertices have different colors?
PERIODIC INDEPENDENT
SET
INPUT: A periodic graph G - (V ,E) and an integer k with 1 < k < IVI.
QUESTION: Is there
set 'S c V such that no two vertices in S are
adjacent and such that S n iP: i
V - k for all p Z?
D/P 3-DIMENSIONAL MATCHING
A collection of triples M c W x X x Y is called periodic if M has the
7
following property: e e M if and only if SUC(e)
INPUT: A periodic collection M c W
x X
M.
x y .
QUESTION: Is there a subset M' c M such that each element of
W
u X u Y appears in exactly one element of M'?
THEOREM 1. The following language recognition problems are all PSPACE-hard:
Bounded D/P Integer Programming, D/P Partition, D/P 3-Satisfiability, Periodic
Hamiltonian Path, Periodic 3-Colorability, Periodic Independent
Set, and D/P
a
3-Dimensional Matching.
We will sketch the proof of Theorem 1 in Section 5. We note that the
first two problems of Theorem 1 are easily shown to be in the class PSPACE.
Without additional conditions it is not clear whether any of the periodic
graph problems listed above or D/P 3-SAT or D/P 3-Dimensional matching is in
class PSPACE.
4. Dynamic Peric tic Languages
In this section we define the class D/P, generalizing several examples of
the previous section. This description parallels one of the methods of
describing the class NP. As such, it is suggestive of the proof technique
given in Section 5. In addition, we show that D/P - PSPACE. For alternate
or [CS] or
descriptions of the class PSPACE, see [GJ
P].
In the following, the symbol 4 will serve as a delimeter. A k-input
language L is a language for which every element has k-I delimeters. The class
kP refers to the subclass of k-input languages that may be recognized in time
polynomial in the size of the first input (the string occurring prior to the
first delimeter.)
For each language L
N(L) - {x: x4y
2P, let
L for some y).
REMARK 1. The class NP is equal to the set of languages
· t - N(L) for some L
such that
2P.
The above remark is a well known alternate characterization of the class
8
NP, For example, Papadimitriou and Steiglitz [PS] use essentially this
characterization as their definition of the class NP.
We will shortly show that a periodic variation of the generation of NP
from 2P can be used to generate the class PSPACE from the class 3P.
For each language L
3P, let D(L) be the dynamic/periodic language
defined as follows:
{x: there exists on infinite sequence [yi
such that
D(L) '
It
+ i
xyiiy
l
e L for each i
i
Z
Z}.
is easy to see that the bounded D/P Integer Programming Problem can be
written so that it
is of the form D(L).
Here the x would stand for the input
and the strings yi would correspond to the solution vectors. We also note that
without boundedness it is unlikely that D/P Integer Programming could be
described in this form. Here the difficulty occurs because the recognition of
the string xyi yi+l must be in time polynomial in
xj.
(That is how we
defined the class 3P.) Thus we could not allow the sizes of the elements yi
and y+l to be exponentially large in
Suppose L
3P and x e D(L).
xi.
We say that a certificate for x is a
sequence {yi: i e Z such that xyi4
yi+l
L for each i
certificate (yi) is periodic with period p if yi
THEOREM 2. Suppose that L e 3P. Then D(L)
Z. We say that a
yi+P for each i
PSPACE. Moreover, there is a
polynomial function q(.) with the following property: for any x
is a periodic certificate (yi) whose period length is at most
PROOF. Because L
true: if xyjy'
Z.
D(L) there
2 q(lxi).
3P, there is a polynomial p(.) for which the following is
L than yl < p(Jxl). Now suppose that x is any string. We
will show how to determine if x
D(L) and, if so, construct a periodic
certificate for x using a standard technique in dynamic programming. Let Gx be
a graph whose vertex set V consists of all strings of size less than p(Ixi).
Let the edge set E consist of all edges (u,v) such that xugv
L. Then
x
D(L) if and only if Gx has a directed circuit. Moreover, the length of the
circuit is at most IVi which is bounded by C P ( lxl), where C-1 is the number of
symbols in the alphabet.
Ill
9
If x
D(L), then we may sequentially guess the periodic certificate in
polynomial space. Thus D(L)
NPSPACE,
the class of languages accepted by non-
deterministic turing machines uing a polynomially space bounded tape. Also
a
NPSPACE - PSPACE, as proved by Savitch [Sal.
Let D/P be the set of all languages
such that
- D(L) for some
L e 3P. We also refer to D/P as the set of dynamic/periodic languages.
THEOREM 3. D/P - PSPACE
PROOF. We have already shown that D/P c PSPACE.
2 is accepted by a polynomially bounded turing machine
Conversely, supose that Z
M. Let p(.) be the polynomial that bounds the space of M.
Let C denote the set of all possible configurations of M with the
restriction that the space is bounded by p([x[). Let L
3P consist of all
triples
x
either
< n,y >
< n2 ,z> satisfying the following:
(i)
nl,n 2 are positive integers less than
Cxl.
(ii)
n2 - nl + 1 and z is the configuration that would follow y
in M,
or else
(ii') n2 - 1, y is'an accepting (terminal) configuration and z is
the initial configuration.
If x
.2,
it is apparent that x
D(L). Moreover, by (ii) and (ii'), any
periodic certificate for x with respect to D(L) corresponds to the sequence of
steps taken by the turing machine M. Thus .
5.
-
o
D(L).
PSPACE-Completeness Proofs
The description of PSPACE in terms of dynamic/periodic language suggests
a range of problems that are candidates for the class PSPACE-complete.
The
obvious first candidate is D/P Satisfiability. Unfortunately, the language as
defined is not necessarily in the class D/P or the class PSPACE. We remedy
this by adding an additional restriction.
We say that a clause c on U
is narrow if any two literals up and v r of
10
the clause are such that Ir-pl < 1. (We could relax the condition so that
Jr-pi
q(JUJ) for some polynomial q. The relaxed condition would still be
restrictive enough for our purposes.) We say that a set C of clauses is narrow
if each clause is narrow. Similarly, a narrow periodic graph is a periodic
graph G - (V",E) such that each edge (iP,jr) is narrow, i.e.,
Ir-p
1.
REMARK 2. If we restrict attention to narrow inputs, the following problems
are all in the class D/P: D/P Satisfiability, D/P 3SAT, Periodic 3Colorability, Periodic Independent Set and Periodic 3-Dimensional matching.
The proof of Remark 2 is straightforward.
Nevertheless it is not always
so straightforward to prove that graph problems on narrow periodic graphs are
in PSPACE. In particular, it
is an open question as to whether the hamiltonian
path problem on narrow periodic graphs is in the class PSPACE. The difficulty
here is due to the fact that the property of being "hamiltoniaan" is not
locally defined, whereas each of the other graph problems listed above is a
local property. For example, to determine if S is an independent set of a
narrow periodic graphs, it
independent for each p
suffices to verify that S n (V P u VP + 1 ) is
Z, where VP is the set of vertices in period p.
A Scheme for Proving the PSPACE-Completeness of Dynamic/Periodic Problems
Suppose that
1
and
B'
are dynamic periodic variants of NP-complete
problems L and L'. Furthermore, suppose thatat is known to be PSPACE-complete.
To prove that'I'is PSPACE complete, first transform L into L' in polynomial
time. Then transforam,
into A' in polynomial time.
This approach might be viewed (albeit,cynically) as the "greedy approach
to proof generation". If 3SAT is used to prove that 3-Colorability is NPcomplete, then one might optimistically hope that D/P 3SAT could be used to
prove the PSPACE-completeness of periodic 3-Colorability. The remarkable thing
is that this greedy approach to PSPACE-completeness proofs usually works.
Intuitively, the description of PSPACE in terms of periodic certificates
is sufficiently close to the description of NP in terms of certificates.Thus
one can prove Theorem 1 by essentially copying the corresponding NPcompleteness proofs from Garey and Johnson [GJ] while keeping careful track of
the superscripts which denote the period. We will omit all the proofs except
III
11
for D/P Partition, whose proof we include for a different reason.
Of course, such a proof technique is not easily formalized, and such a
mimicking does not exist in general. Moreover, the "mapping" of NP problems
into D/P problems is not clearly defined. For example, the periodic
counterpart to the hamiltonian circuit problem may not be the D/P hamiltonian
path problem. Maybe the correct counterpart is the problem finding a subgraph
S such that when restricted to the vertex set VP u VP + 1, S is a hamiltonian
circuit.
The above definition is the "natural" generalization in terms of the
class D/P, but it is less natural from a graph theoretic point of view. The
D/P generalization of the clique problem is just as unnatural from a graph
theoretic point of view, viz., select a set S such that S n (VP u VP+1 ) is a
clique for all p c Z.
Although the greedy approach to PSPACE-completeness proofs works usually,
we illustrate below a case in which an added degree of subtlety is
required.
In the remainder of this section we show how to transform the D/P knapsack
problem into the D/P partition problem.
KNAPSACK
INPUT: non-negative integer n-vector a, and integer d.
QUESTION: Is there as 0-1 n-vector x such that ax - d?
D/P KNAPSACK
INPUT: non-negative integer n-vectors a and b, and integer d.
QUESTION: Is there an infinite sequence x - {xi: i
i+1
i
such that ax
+ bx - d for all i
Z?
Z} of 0-1 n-vectors
The partition problem is the special case of the knapsack problem in which d =
la/2.
The usual polynomial time transformation from the knapsack problem into
the partition problem is straightforward. Let a and d be the input for the
knapsack problem. Let a' be the (n+l)-vector
a,al where a'
- 12d-lal, and let d' - la'. Then there is a 0-1 vector x
n+1
r+1
such that ax - d if and only if there is a 0-1 vector x' such that a'x' - d'.
The validity of the "if" direction depends on the following simple
12
observation: if a'x'
d',
then
'(l-x') - d'. The solution to the knapsack
problem is either x'
or l-x' according as xt
n+ 1 - 0 or x'
n+1 - 1.
The same argument does not apply to the D/P knapsack problem. Suppose one
were to attempt to transform the D/P knapsack problem with inputs a,b,d into
the D/P partition problem with input (a',b',d') defined as follows:
a' - (a,a;+l), b' - (b,O) and al
If there is
-
2d-la-lbl, and d' - (la'+lb')/2.
a solution to the D/P knapsack problem then this is a solution to
the D/P partition problem, but the converse is not true. A simple example is
the problem "3xi + 4x
+l
Nevertheless, the problem
* 4", which admits no periodic solution.
3x
+ yi + 4x+1 _
4" does have a periodic
solution.
There are two remedies: change the problem or change the transformation.
The first remedy is actually not frivolous, since it raises a relevant issue.
Perhaps the appropriate formulation of knapsack and D/P knapsack are as
follows.
KNAPSACK II
INPUT: non-negative integer n-vector a, and integer d.
QUESTION: Is there a 0-1 n-vector x such that either ax - d or else
ax - (la-d)?
D/P KNAPSACK II
INPUT: non-negative integer n-vectors a and b, and integer d.
QUESTION: Is there an infinite sequence x such that
i+ l
ax
+ bxi - d or la + lb - d
{xi: i
for each i
Z
of 0-1 n-vectors
Z?
Knapsack and knapsack II are the same problem. However, D/P knapsack II
is different from D/P knapsack; moreover, the above transformation shows that
D/P Partition is PSPACE-complete assuming that D/P knapsack II is PSPACEcomplete. The point is that the failure of the mimicking of the NPcompleteness proof may be due in part to the ambiguity in how one should
define D/P knapsack.
To transform D/P knapsack into D/P partition is not difficult, but it is
more difficult that the corresponding NP-complete transformation. Let a,b,d be
the input for D/P knapsack. Let a' - ala
2 and let b' - b,bl,b 1l, where
,an1'+
2
III
13
b'
-M + 2d - la - lb
n+l
'
- 1
n+2
d' - (la'+lb')/2,
where M is greater than la + lb. The reader can verify that the transformation
is correct. The key difference between this transformation and the previous
one is that in any feasible solution to the D/P Partition problem we must have
for each p Z.
and
that xP
n2
n+2
n+1
n+1
6. A Pseudo-Polynomial Algorithm for the D/P knapsack Optimization Problem
The following rules of thumb appear to be widely applicable. If a
language L is NP-complete then the analogous dynamic/periodic language is
PSPACE complete. If a language L is in the class P, then the analogous
dynamic/periodic language is also in the class P. Examples of dynamic/periodic
problems soluable in polynomial time may be found in Graves and Orlin [GO] and
in Orlin [02], [03], [041, [05] and 06]. Although the above rules of thumb
are not true in general, they do appear to be true for many "natural"
problems.
The above rules of thumb suggest that the D/P knapsack problem should be
soluable in pseudo-polynomial time and should be PSPACE-complete. In fact,
both properties are true. Below we will show how to solve the following
optimization version of the D/P knapsack problem in pseudo-polynomial time.
Minimize p
cx
i-l
Subject
to
ax
ax
i
p
O <
+ bx
+ bx
xi
i+l
d
for i - 1,...,p-
= d
< 1, x i integer
p > 0 integer.
Here, for convenience, we have written a periodic description in which the
period length p is a variable.
To solve the problem, we create a directed graph G(a,b,c,d) - (V,E),
14
where V - {O,...,d). Moreover, tere is
there is a 0-1 n-vector x such that ax -
an edge (i,j) with cost k in E if
, bx - d-i, and cx - k. The graph
may be constructed in O(nd2) steps using a dynamic programming recursion.
THEOREM 4. There is a feasible solution to the D/P knapsack problem if and
only if G(a,b,c,d) has a directed circuit. The optimum solution, if one
exists, is induced by the circuit C in G which minimizes the ratio of the cost
to the number of edges. (This ratio is called the minimum cycle mean).
PROOF. Let C be a directed circuit in G with edges el,...,e
p.
Let xl,...,xP be
the 0-1 n-vector that induced the edge. Then xl,...,xP in a periodic solution.
Moreover the average cost of the circuit C is pl(cxl+...+cxP), which is
the
cycle mean of C. It is easy to verify that the correspondence between circuits
of G and periodic solutions is
1:1.
We note that the minimum cycle mean of G may be determined in O(d3 ) steps
using an algorithm of Karp [K].
7. Real time scheduling
In the dynamic/periodic optimization problems described in Section 2, the
schedule (certificate) is the crucial output to obtain. Moreover, even if one
had an oracle to solve the language recognition problem, there is
no apparent
polynomial algorithm to find the certificate. Indeed, the certificate will
often be exponentially long.
In most of the scheduling problems of Section 2, the schedule is
to run
over an infinite horizon. Generally, it is unnecessary or unimportant to
obtain the infinite-horizon schedule in polynomial time. It suffices to
determine the schedule for the first i periods in time polynomial in the data
and in i.
We might view such an algorithm as a polynomial real time algorithm.
We will show below that none of the D/P problems of Theorem 1 have
polynomial real time algorithms, unless P - PSPACE.
just to determine the certificate for period 1 is
In fact, we will show that
PSPACE-hard.
Let L c 3P. The initialization problem for D(L) is as follows.
15
INPUT: y
D(L)
PROBLEM: Determine xI such that there is a periodic certificate
(xi: i
1 ,...,p).
THEOREM 5. The initialization problem is PSPACE-hard for all of the D/P
problems in Theorem 1.
SKETCH OF PROOF. Modify the proof that D/P satisfiability is PSPACE-complete
as follows. Starting with an arbitrary language A
transform x to an instance x' such that x'
PSPACE and an instance x,
D/P Satisfiability, and any
certificate (yi) for x' has the following property
(i)
The first symbol of yi is 0 for all i if x
(ii)
The first symbol of yi is
1 for all i if
.
x I.
The details of the transformation are detailed but straightforward. Also, if
yi is
part of a valid certificate for x',
one can determine whether x
thus the initialization problem for D/P satisfiability is PSPACE-hard.
, and
The
other proofs are analogous.
We note that in our proof that D/P
PSPACE, all the PSPACE languages
I- D(L) so constructed are polynomially real time solvable.
8. Extensions
Suppose that L
3P and S,T
P, and t
Z. One can consider variations
of the language D(L) as follows.
FIXED PERIOD PROBLEM
INPUT: x,t
QUESTION: Is there a periodic certificate (yi: i-l,...,t) for x?
INITIAL CONDITION PROBLEM
INPUT: x,S
QUESTION: Is there a certificate (yi: i
+
(i.e., xyi+yi l
L for i > 1 and yl
S).
Z+ ) for x such that yl
S.
16
INITIAL AND TERMINAL CONDITION PROBLEM
INPUT: s,S,T
QUESTION: Is there a certificate yl,..,yt such that xtyiyi+l e L
for i
1,...,t-1 and yl¢
S and yt
T.
One may also form the latter problem with a fixed period length. As long as
the period length is
not bounded, all of the PSPACE-completeness Theorems of
this paper extend to the above generalizations.
9. Open Problems
In order to prove that the periodic graph problems are in the class
PSPACE, we restricted attention to narrow periodic graphs. It is
an
interesting open question as to whether the non-narrow periodic graph problems
are in the class PSPACE. (This seems quite unlikely.) If not, they may form an
interesting class of problems that provably require an exponential amount of
space. Also, the complexity of the periodic hamiltonian path problem on narrow
periodic graphs is of interest. (We know that it is PSPACE-hard.)
On a broader scope, it is of interest to know if there are general
properties that an NP-complete problem may have so as to guarantee that the
corresponding D/P problem is PSPACE-complete. Currently, we have no metatheorems concerning these PSPACE-completeness results.
Acknowledgements
I wish to thank Christos Papadimitriou for several helpful discussions.
This work was supported in part by the National Science Foundation.
17
PERIOD
1
Vertex
3
3
2
3
4
5
. . .
.
.
. . .
.
.
. .
Figure 1. A subgraph of a (narrow) periodic graph.
18
References
[B]
Baker, K.R., "Workforce Allocation in Cyclical Scheduling Problems",
Operational Research Quarterly, 27, (1976), pp.155-167.
[C]
Cook, S.A., "The Complexity of Theorem-Proving Procedures", Proc. 3rd ACM
Symposium on The Theory of Computing, New York, (1971), pp.151-158.
[CS] Chandra, A.K. and L.
FOCS, (1976),
[D]
Dantzig,
"Alternation",
Stockmeyer,
Proceedings of the 17th
pp.98-108.
G.B., W. Blattner, and M.R.
Rao, "Finding a Cycle in a Graph
with Minimum Cost to Times Ratio with Application to a Ship Routing
Problem",
in Theory of Graphs,
Gordon and Breach, New York,
[DL] Dhall,
S.K.,
and C.L. Liu,
Operations Research,
[FF]
26,
P.
Rosentiehl, editor, Dinot, Paris;
(1967),
pp.77-84.
"On a Real-Time Scheduling Problem",
(1978),
Ford, L.R., and D.R. Fulkerson,
pp.127-140.
Flows in Networks,
Princeton University
Press, Princeton, N.S., 1962.
[GJ] Garey, M.R., and D.S. Johnson, Computers and Intractability: A Guide to
the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco,
1979.
[GO] Graves, S.C., and J.B. Orlin,"A Minimum Concave Cost Dynamic Network Flow
Problem with an Application to Lot-Sizing", Networks 15, (1985), 59-71.
[K]
Karp, R.M., "Characterization of the Minimum Cycle Mean in a Digraph",
Discrete Mathematics 23, (1978), pp.309-311.
[L]
Lawler, E.L., "Optimal Cycles in Doubly Weighted Linear Graphs", in
Theory of Graphs, P. Rosentiehl, editor, Dinod, Paris; Gordon and Breach,
New York, (1967), pp.209-214.
[La] Labetoulle, J. "Some Theorems on Real-Time Scheduling", Computer
Architecture and Networks, E. Gelenbe and R. Mahl editors, North Holland
Publishing Company, (1974), pp. 285-298.
[LL] Liu, C.L., and J.W. Layland, "Scheduling Algorithms for Multiprogramming
in a Hard Real-Time Environment", JACM, 20, (1973), pp.46-61.
[LM] Lawler, E.L. and C.U. Martel, "Scheduling Periodically Occurring Tasks on
Multiple Processors, submitted for publication.
[M]
Megiddo, N., "Combinatorial Optimization with Rational Objective
Functions", in Proceedings of the 10th ACM Symposium on Computation, San
Diego, California, (1978), pp.1-12.
III
19
[Ma] Periodic Optimization Vol. Z, A. Marzollo editor, CISM Courses and
Lectures 135, Springer Verlag, 1972.
[Mo] Mok, A., "Complexity of Static Schedule Design for Hard Real-time
Environments", Working Paper, Laboratory for Computer Science, MIT, 1980.
[011 Orlin, J.B., "The Complexity of Dynamic Languages and Dynamic
Optimization Problems", Transactions of the 13th Annual ACM Symposium on
the Theory of Computing, May 1981, pp.218-227.
[02] Orlin, J.B.,"Minimizing the Number of Vehicles to Meet a Fixed Periodic
Schedule: an Application of Periodic Posets", Operations Research 30,
(1982), pp.768-776.
[03] Orlin, J.B., "Maximum Throughput Dynamic Network Flows", Mathematical
Programming 27, (1983), pp.214-223.
[04] Orlin, J.B., "Dynamic Matchings and Quasi-dynamic Fractions Matchings I
and II", Networks 13, (1983), pp. 551-580.
[05] Orlin, J.B.,"Minimum Convex Cost Dynamic Network Flows", Mathematics of
Operations Research 9, (1984), pp.190-207.
[06] Orlin, J.B., "Algorithms for Problems on Dynamic/Periodic Graphs", in
Progress in Combinatorial Optimization, William Pulleyblank, ed.,
Academic Press, Toronto, 1984.
[P]
Papadimitriou, C.A., "Games Against Nature", Proceedings of the 24th
FOCS, (1983), pp.446-450.
[PS] Papadimitriou, C.A. and K. Steiglitz, Combinatorial Optimization:
Algorithms and Complexity, Prentice-Hall, 1981.
[R]
Reif, J.H., "Complexity of the Movers Problem and Generalizations",
Proceedings of the 20th FOCS, (1979), pp.421-427.
[Sa] Savitch, W.J.,"Relationship Between Non-Deterministic and Deterministic
Tape Complexities", J. Comput. Systems Science 4, (1970), pp.177-192.
[Sc] Schaefer, T., "The Complexity of Some Two-Person Perfect Information
Games", J. Comput. Systems Science, 16, (1978), pp. 185-225.
[Si] Simpson, R.W., "Scheduling and Routing Models for Airline Systems",
Technical Report 1268-3, Flight Transportation Laboratory, MIT, 1969.